Simplify the following expression and state the condition under which the simplification is valid. You can assume that $t \neq 0$. $r = \dfrac{4}{5(4t + 9)} \div \dfrac{3t}{t(4t + 9)} $
Answer: Dividing by an expression is the same as multiplying by its inverse. $r = \dfrac{4}{5(4t + 9)} \times \dfrac{t(4t + 9)}{3t} $ When multiplying fractions, we multiply the numerators and the denominators. $r = \dfrac{ 4 \times t(4t + 9) } { 5(4t + 9) \times 3t } $ $ r = \dfrac{4t(4t + 9)}{15t(4t + 9)} $ We can cancel the $4t + 9$ so long as $4t + 9 \neq 0$ Therefore $t \neq -\dfrac{9}{4}$ $r = \dfrac{4t \cancel{(4t + 9})}{15t \cancel{(4t + 9)}} = \dfrac{4t}{15t} = \dfrac{4}{15} $